Believe those who are seeking the truth. Doubt those who find it. Andre Gide

Sunday, December 1, 2013

Is QE lowering the rate of inflation?

The answer may be "yes," according to a new paper by Steve Williamson. In examining the effects of a QE experiment in his model economy, he reports the following (p. 16):

Some of the effects here are unconventional. While the decline in nominal bond yields looks like the "monetary easing" associated with an open market purchase, the reduction in real bond yields that comes with this is permanent, and the inflation rate declines permanently. Conventionally-studied channels for monetary easing typically work through temporary declines in real interest rates and increases in the inflation rate. What is going on here? The change in monetary policy that occurs here is a permanent increase in the size of the central bank's holdings of short-maturity government debt - in real terms - which must be balanced by an increase in the real quantity of currency held by the public. To induce people to hold more currency, its return must rise, so the inflation rate must fall. In turn, this produces a negative Fisher effect on nominal bond yields, and real rates fall because of a decline in the quantity of eligible collateral outstanding, i.e. short maturity debt has been transferred from the private sector to the central bank.

So what happened? It all started out by Williamson discussing two "fearsome equations" that emerge as theoretical restrictions in a wide class of macroeconomic models. Some people call these restrictions "Fisher equations." I like to think of them as no-arbitrage-conditions.

Let's make some assumptions. There is no uncertainty. We are (the model economy is) in a state-state, so that output grows at the gross rate G. The level of output may be above or below its "natural" level (the level that would prevail if all frictions were absent). Let B denote the discount factor. Absent all frictions, the "natural" rate of interest is given by (G/B).

Let P denote the gross rate of inflation. There are two nominal assets, a bond that yields a gross nominal return R >= 1, and money, which yields a gross nominal return equal to 1. Money is assumed to be more liquid than bonds (bonds cannot be used in a subset of transactions).

The "Fisher equations" that emerge from the model can be written as follows:

[1] R*K = (G/B)*P and [2] 1*L = (G/B)*P

where K and L denote "liquidity premia." In most models, K=1. In this case, the two equations above imply R = L. That is, the liquidity premium on money is equal to the nominal interest rate.

In the "newmonetarist" models that Steve studies, assets apart from money may serve in some manner as exchange media. If financial markets do not work perfectly well (say, because of limited commitment and asymmetric information frictions), then the supply of exchange media may be "scarce." In the present context, this implies K>1, and equations [1] and [2] imply: R*K = L.

The traditional Friedman rule policy implies R = 1, K = L =1, so that P = (B/G). But Steve is assuming the government does not have enough instruments to implement the Friedman rule. In fact, he makes a distinction between the monetary and fiscal authorities. And, as he stresses in his paper (not his blog post), a lot hinges on exactly how one models this relationship.

One scenario that emerges in Steve's model is R =1 and K = L > 1. In this case, the economy is at the ZLB, but government liabilities (cash and bonds -- they are perfect substitutes in this case) exhibit a liquidity premium. On open market operation of cash for bonds in this case has absolutely no effect -- this is the classic liquidity trap -- something that Krugman stressed long ago. From [1] and [2], the equilibrium inflation rate is given by P = (B/G). The equilibrium real rate of interest on government liabilities is 1/P = G/(B*L), which is less than the "natural" real rate of interest (G/B). This is the sense in which the real rate of interest is "too low." (Of course, if you have a different theory of the way the world works, you may be thinking that the real rate of interest is "too high"--but I'm not here to talk about that theory.)

Suppose that the bond we are talking about above is a short-maturity instrument. Imagine that the fiscal authority also issues a long-maturity instrument. Moreover, assume that this long-bond is less liquid than the short-bond (the short-bond is, in present circumstances, viewed as a perfect substitute for cash). Steve then asks what the model implies when the open market operation consists of a swap of cash for the long-bond. In this case, not surprisingly, QE matters. But how does it matter?

The effect of this policy in Williamson's model is to lower the nominal interest rate at the long end of the term structure. Because the Fed is sucking out relatively less liquid assets and replacing them with relatively liquid assets, liquidity premia decline (as one would expect). So, if we take a look at equation [2], we see that the model implies that inflation must decline: P = (B*L)/G. What is the economic intuitions for this? Evidently, one of the effects of QE (in the model) is to increase the real stock of currency held by the private sector, and agents require an increase in currency's rate of return (a fall in the inflation rate) to induce them to hold more currency. (Remember that the results are all contingent on the way monetary and fiscal policy are modeled.)

So this is kind of interesting for a couple of reasons. First, the model offers an explanation for why we do not observe deflation, given that we are at the ZLB (a bit of a puzzle, for conventional theory.) Second, it offers an explanation for how QE may be putting downward pressure on inflation. How quantitatively important these effects are relative to others remains an open question.

Krugman and DeLong seem to want to argue that Williamson's results are "incorrect" because the model equilibrium he is focusing on is "unstable." I'm pretty sure I know where they're coming from, but I'm not sure that the criticism applies here.

First, to demonstrate the "stability properties" on an equilibrium, one actually has to go and work out the math. I think it's fair to say that nobody has done that.

Second, what Krugman writes in his "Little Arrows" post is correct, but it is correct only in the context of a particular theory. As I've mentioned before, Peter Howitt demonstrates here how pegging the nominal interest rate is unstable under a wide class of algorithms that govern the manner in which inflation expectations are formed (essentially, adaptive expectations). This led Howitt to argue that stability required a policy to raise interest rates more than one-for-one with inflation expectations. Hence, Howitt came up with the "Taylor principle" before Taylor did.

It is interesting to note, however, that the "stability properties" induced by the Taylor principle in standard New Keynesian models (which embed rational expectations) is something very different. At the opposite extreme, one might take the view that inflation expectations are formed in a manner described here, by Stephanie Schmitt-Grohe and Martin Uribe. Let me reproduce the diagram I used in that blog post here:

As you can see, it is identical to Krugman's "Little Arrows" diagram. The one big difference here is that--under this particular theory of expectations formation--rational expectations--A is unstable and B is stable. So the "little arrows" run in the opposite direction here.

The little circle in the picture above demonstrates how the New Keynesians use the "Taylor principle." Essentially, they restrict attention to trajectories around the steady state point A that never leave that circle. If the Fed follows the Taylor principle, then there is only one point that satisfies this property, and it is point A. Viola-we say that point A is "locally stable" (Yes, I know it sounds weird, but I'm just reporting the facts.) In their Perils of Taylor Rules, Benhabib, Schmitt-Grohe, and Uribe argue that only point B (the liquidity trap) is globally stable. (The fact that Japan has spent decades around a point like B suggests that it may in fact be stable.)

The other thing I'd like to add is that Williamson's results continue to hold even away from the ZLB. So, Krugman's post in particular, which focuses on the properties of Taylor rules (absent in Williamson's model) seems a little off target.

So my interpretation of the criticisms I am hearing of Williamson's paper is that his critics are claiming that he is wrong because his results are inconsistent with the type of models these people are used to working with. It seems to me that the critics should have instead attacked his results and interpretations with empirical facts (or am I too old-fashioned in this regard?). After all, Williamson at least motivated his post with some data (the diagram at the top of this post). And he makes what is potentially a testable prediction (notice the if-then structure of the statement):

In general, if we think that inflation is being driven by the liquidity premium on government debt at the zero lower bound, then if the Fed keeps the interest rate on reserves where it is for an extended period of time, we should expect less inflation rather than more.

I have a little more difficulty in understanding Nick Rowe's objection. Certainly, a part of it seems to be what I just described above. Partly, I think that Nick is disagreeing not with Williamson's model, but with the way Williamson seems to run off at the end of his post with his "the Fed is in a trap" ideas.

And so, this now leads me to my own criticism of Williamson's post.

I wish he had spent a little more time elaborating on this statement he makes:

But the power of monetary policy to mitigate the inefficiency is limited. Basically, it's a fiscal problem. The U.S. government could issue more debt, by temporarily running a higher deficit. But that's not happening, so what can the central bank do about it?

It's a fiscal problem (well, the fundamental problem is limited commitment and asymmetric information in these models). The Treasury could alleviate the "asset shortage" by expanding the supply of Treasury debt! I discussed this idea here some time ago: Not Enough Debt? So isn't this nice? Different models, but similar policy conclusions. Implicitly, Williamson is taking the view that political constraints are preventing this from happening, so let's move on to study Fed policy.

The tone of his post near the end strikes me as odd. He seems rather critical of the way Fed economists generally think about the way monetary policy works. Fair enough. On the other hand, if we read his paper we find the following statement:

QE is a good thing, as purchases of long-maturity government debt by the central bank will always increase the value of the stock of collateralizable wealth.

That is, QE is a good thing in his model economy. In fact, I think his model suggests that the Fed should buy up all outstanding treasury debt (but that even that would not be enough because the problem is the limited supply of the stuff).

So what's his problem? Well, it seems that conventional Fed thinking is that QE is inflationary and, well, as Williamson's paper shows, it may have the opposite effect. O.K., well, so what?

Then Williamson remarks that if the Fed really wants inflation, it should raise its policy rate (IOER). This, of course, is the statement that drew all sorts of criticism when Narayana Kocherlakota suggested something similar a few years back (thanks to Nick Rowe for once again starting that one). Williamson believes that raising the policy rate would be disruptive in the short-run, but that this is the way to achieve higher inflation in the long-run. I am not sure, however, whether his model suggests that higher inflation is a good thing (I don't think so.) These are all positive (not normative) statements.

So, the Fed is "stuck." That is, the Fed seems compelled to continue QE and keep the IOER at 0.25%. Williamson's model seems to suggest this is a good thing. But his model also suggests that the policy is ultimately deflationary (perceived to be a bad thing). The only way to prevent this trajectory is to raise the IOER (another way would be to expand the supply of treasury debt). But doing so will cause a recession because of monetary non-neutralities.

Not sure what any of this has to do with eating more crow though. What would the Fed be doing differently if they took this view? Not much, as far as I can see.

From the Benhabib/Schmitt-Grohe/Uribe working paper (I couldn't download the other):

"Since the inflation rate is a jump variable whose initial value is not exogenouslygiven, the existence of a steady state with active policies immediatelyestablishes the possibility of the existence of at least two steady state equilibria."

That's the key. And what they really mean is: "Since *expected* inflation is a jump variable..."

Now let me translate it all into English:

Think of the conductor of an orchestra, or the coxswain on a racing eight, calling out a Schelling focal point for expected inflation.

Start in full-employment equilibrium. Then the central bank suddenly raises the nominal interest rate 1%, for no reason at all.

The conductor, if he is omniscient, now announces: "Let us all now expect 1% higher inflation, because if we don't all expect higher inflation, the Euler equation will not be satisfied in continuous full-employment!"

But what happens if they don't hear the conductor? Maybe each (atomistic) agent thinks that every other agent's rate of time preference has increased by 1%, so the central bank needed to raise the nominal rate by 1%, just to keep the same equilibrium rate of inflation. So each agent expects the same rate of inflation going forward.

What *would* happen, counterfactually, if the Euler equation were not satisfied at full-employment?

NK models assume (for no reason at all, but let that pass), that all agents expect the economy will eventually return to full-employment. Given that assumption, if the real rate increases, then the Euler equation says that current output will be less than full-employment output. There would be deficient aggregate demand in the current period.

So, if agents do not hear the conductor, and do not change their expectations of inflation, each discovers there is excess supply of his output. What happens next? Under any plausible story I can think of, each individual agent will plan to lower his relative price, which means actual inflation will fall below expected inflation.

Who is the conductor? The only plausible candidate is the central bank itself. If the central bank announces an increase in the target rate of inflation and an increase in the nominal interest rate, both at the same time, the result would be very different than if it did the latter without the former.

Note: the agents in my story do not have irrational expectations. But they do have imperfect information. They do not know *why* the central bank raised the nominal interest rate. It matters.

Suppose the natural rate of interest falls. But the representative agent does not know it has fallen. (The representative agent does not know he is the representative agent, so he knows that his own rate of time preference has fallen, but does not know that every other agent's rate has fallen too.)

And suppose the central bank does not cut the nominal interest rate (perhaps it cannot).

The representative agent does not change his expected rate of inflation, since he thinks the shock is specific to him. So he cuts his consumption, and is surprised to see his customers doing the same. So he increases his price by less than his expectation of inflation. So actual inflation falls.

There are two totally different notions of stability being used in this debate as if they were in fact equivalent. When Howitt and Evans/Honkapohja talk about stability they mean stability of a steady state with respect to *disequilibrium* dynamics or learning, in which expectations are mutually inconsistent except when the steady state is reached. When Benhabib et al. talk about stability they are talking about convergence of *equilibrium trajectories* to steady states, so expectations satisfy RE even outside the steady state.

These are not alternative theories of expectation formation: one is a theory of expectation formation and the other is the imposition of an equilibrium condition. One deals with how expectational consistency might arise, the other simply assumes it. It's misleading to threat these two approaches as just two different theories of expectation formation.

Here's Woodford from a couple of years in response to John Kay:

"The macroeconomics of the future... will have to go beyond conventional late-twentieth-century methodology... by making the formation and revision of expectations an object of analysis in its own right, rather than treating this as something that should already be uniquely determined once the other elements of an economic model (specifications of preferences, technology, market structure, and government policies) have been settled."

A solution concept is a method not a theory. Nash equilibrium for instance is not a theory of expectations, it's a solution concept based on an assumption that expectations are consistent. RE is no different.

The reason why DeLong keeps going on about Frankin Fisher is that Fisher was adamant that equilibrium concepts had to have disequilibrium foundations. This was a really active area of research in general equilibrium theory in the 60s but faded away because no general stability results could be obtained. Fisher was one of the few who kept going. I think Woodford is arguing that we need to bring this focus back.

Meanwhile, it's fair to evaluate an RE equilibrium path (steady state or otherwise) based on whether one thinks there's a plausible mechanism for reaching it in a decentralized economy. The Howitt paper (which you first brought to my attention with a comment on Steve's blog) does this for one type of RE model. I don't think anyone has done this for the Benhabib et al. model.

I'm a bit obsessed with these issues so I keep bringing them up on my blog (when time permits). I just think that treating AE and RE as just two different models of expectations is wrong and leads to a lot of confusion once we get into questions of stability.

Can you please provide me your definitions of "solution concept," "method," and "theory." Once you have done this, I can more easily assess your claim. (Btw, I have a sense of what you are saying, I just want you to make it as clear as possible -- thanks!)

@ David, why do you care what "solution concept" or "method" mean to Rajiv, you don't need them to understand the gist, stay focused!

@ both, my reading of Paul Krugman is this: it's acceptable to focus on equilibrium paths if there are plausible mechanisms that suggest they might be reached. And on this, I'd like to point you all to the seminal, underappreciated and underquoted work of one Steven Goldman in the (I think) early 1980s: he anticipated the sort of nearly-rational short-horizon semi-myopic forward-looking stuff that Mankiw and others have made popular recently. Ricardo Reis has some absolutely stunning papers on that, he may be the one who revived it.

So PK would have a preference for the fully fledged Evans-Howitt-Sargent, etc. tatonnement type out-of-equilibrium mechanisms, but since it's a pain to model and has a zillion solutions concepts, in the meantime he'll ask for a plausible story.

Is a plausible story forthcoming chaps?

@David. Thanks for an active discussion, it's always healthy to have. But please tone down your self-confidence. You don't want to sound like Paul Krugman or Stephen Williamson do you?

David, I'll give this a shot though a philosopher of science would do a better job. To my mind a theory in social science is an explanation or account of certain observed phenomena based on clearly articulated behavioral hypothesies. So, for example, both Duesenberry and Friedman developed theories of saving to account for the (then) puzzling fact that the rich appeared to save a greater potion of their current income than the poor in cross sectional data, but countries did not seem to save more as the became richer in time series data. Duesenberry emphasized relative income comparisons, Friedman focused on permanent versus temporary shocks to income, but both were well articulated theories.

Similarly, the adaptive expectations hypothesis is a theory of expectation revision, which specifies how prior expectations and realizations of a variable combine to determine current expectations. It's crude, and one can get much more sophisticated by treating agents as econometricians with models in their heads that are estimated using realized data along lines proposed by Marcet and Sargent, or Howitt, or Evens and Honkaphja, or Friedman and Phelps. These are all theories of expectation formation, or revision, or calculation: they tell you how expectations and realizations combine to give you revised expectations.

RE is different. It just specifies that individuals hold precisely those expectations that, if held, would be self-fulfilling. It does not specify any process, either cognitive or based on real time learning, that would make such a hypothesis plausible. It's an equilibrium hypothesis, based on the implicit but rarely articulated belief that the equilibrium in question is stable with respect to some plausible cognitive or learning process in a decentralized economy.

In game theory a bit more thought is given to reasoning and disequilibrium dynamics. For example, the notion of stochastic stability leads to the selection of a particular equilibrium in generic coordination games. Equilibria based on level-k reasoning are justified by a well articulated cognitive process. And sampling dynamics can be used to select among sampling equilibria. It seems that more attention to stability as a source of equilibrium (or disequilibrium) selection would be helpful in Macro.

Excellent job Rajiv. I think it's fair to say that game theory has shown that you can't really have rationality in the sense of RE AND disequilibrium dynamics in the sense of out-of-equilibrium path beliefs (or something) because you get caught in an infinity of infinities, hence level-k reasoning with k finite and quite small.

Rajiv, thank you (btw, I was looking for something you may have posted on your blog concerning this topic, but evidently you did not?)

I appreciate the point you are making. In my view, this is all a very difficult business. At some point, we need to impose a solution concept. Having imposed that solution concept, there will always be another layer of hand-waving or story-telling about how "we got there." So, my personal preference is to remain eclectic on the matter. Let people construct theories they way they think is useful, then let them try to interpret the data, make predictions, and persuade us that the interpretation is useful.

@Patrick, go take a look at Williamson's blog if you are interested in story telling. I am a bit surprised with Krugman on this front. He frequently cites James Tobin's general equilibrium theory of money as a way to organize thinking about inflation and financial markets. Steve is working in that tradition. I wish that Krugman was more interested in building bridges, rather than canals.

David, I agree that we need to be eclectic, and the learning literature is difficult and highly specialized. We can't have disequilibrium analysis in every equilibrium model. But when one gets comparative statics results in an equilibrium model that defy common sense and economic intuition, I think it's important to check for robustness.

By the way I also endorse your point about building bridges, and wish that Krugman would be less aggressive in making his points. But I also find the tone on Williamson's blog to be insufferably smug and condescending. Nick Rowe is a model of civility in comparison... seems to be a really good natured guy. Would like to meet him some day.

@ David, I'll be frank I wasn't able to read much of Williamson's blog, I found the tone far too unpleasant. But fair point, you're right, I "should" fish his blog for stories, it's just that I probably won't.

From your old post: "Start with an inflation rate close to but below target. Then trace your pencil up to the Taylor Rule line--this is the policy rate associated with the initial inflation rate. Now ask: at this policy rate, what does the Euler Equation imply about the expected (one period ahead) inflation rate?"

No! Again, that tells us what expected inflation *would need to be* to keep output at full employment. It does not tell us what expected inflation would actually be.

This is what sociologists call the Functionalist Fallacy. Yes, it would be good if all agents expected that, but why would any individual agent choose to expect that?

Nick, I have no idea why agents would expect that. I also have no idea how my dog catches the frisbee I throw him. It is "as if" the mutt is performing some complicated mathematical operation (I know you are familiar with this argument -- it is from Friedman). By making the assumption that my dog somehow solves this complicated problem, I can make predictions. And ultimately, our theories will have to be consistent with the data.

Deep theories of expectation formation are hard (at least, I find them difficult). Adaptive expectations seem to work well in many circumstances. But not always; see Tom Sargent here:

David: OK. But I assume my dog would catch the frisbee in the individually optimal way, not the socially optimal way. What you are assuming here is that individual agents choose the socially optimal inflation rate to ensure full employment.

Btw, you are wrong to say that agents are assumed to coordinate in a socially optimal way. The equilibrium they coordinate to in Steve's model is generically socially inefficient. The implausible idea, which most people have trouble with, is the assumed coordination. But that's the nature of the solution concept. People are free to invent their own solution concepts and see where it gets them.

David: "Btw, you are wrong to say that agents are assumed to coordinate in a socially optimal way. The equilibrium they coordinate to in Steve's model is generically socially inefficient."

Understood. Because the Friedman rule is not generally satisfied. I was ignoring that suboptimality, and referring only to the full-employment condition.

But then why shouldn't Steve assume agents coordinate on some second-best optimal convex combination of full-employment and the Friedman Rule? An Old Keynesian would complain that Steve is *just assuming* full employment, and would be right to make that complaint. Normally we want a model in which it is individually optimal for individuals to take actions that ensure full-employment.

When you apply Williamson's model outside of a liquidity trap, doesn't it say the CB will increase the inflation rate when it raises interest rates? There the CB is just increasing the liquidity premium on money as opposed to money/bonds when it raises rates through conventional OMOs, which I think would raise inflation if you used the same fixed-value-of-government-debt passive fiscal policy and price determination assumptions employed in the model. Even without any kind of plausible learning mechanism, that story looks more like frisbees chasing dogs compared to what we actually see in the world. What is supposed to change in the liquidity trap?

In the version in the Liquidity Trap notes (I figured the important nitty gritty was in the linked pdfs, so actually I only skimmed the blog post itself)

http://www.artsci.wustl.edu/~swilliam/papers/liquiditytraps.pdf

the effect of a change in the interest rate when not in a liquidity trap depends on elasticity int sub/risk aversion.

In that model inflation depends positively on the marginal utility of the money-constrained good. A cut in the interest rate changes the government's (and agent's) budget constraint but also changes the marginal rate of substitution between the money-constrained good and the bonds-and-money-constrained good. This means that consumption of money constrained good can go up or down depending on which effect dominates and so can inflation.

Actually come to think of it, this is just like old Tobin's Portfolio Balance model of money demand, with liquidity instead of risk. There too there is a substitution and income effect and depending on how it goes, money demand can be positively or negatively related to the interest rate. Of course everyone believes the inverse relationship but that's for empirical reasons.

At first, I agreed with Krugman, but as I took on his challenge to explain why agents would behave this way, Williamson's idea seems to make sense. I've posted several comments under Krugman's post that offer different ways of viewing how the disinflation outcome makes narrative sense (see J. Hansen comments).

Perhaps a distilled view is simply that in the modern economy, most agents prefer to be paid more rather than lower their prices (downward nominal rigidity restated), and this effect plays out at the firm level in the form of happily increasing the firm's defacto time-preference by refinancing to much longer-term debt rather than cutting prices or layoffs of remaining employees. QE enables this shift, and the effect is to subvert firm's impetus to raise prices to meet those previous interest payments. As long as low-interest refi options are available, there is little reason to raise prices unless demand seems available.

This view hinges on an economy where firms are deficit financed. Another implicit assumption is long-lived agents - such as corporations. Predominantly short(er)-lived agents (partnerships and humans) pay a higher premium, so the effect is smaller. Long-lived corps can obtain indefinitely low rates - low risk premiums. Thus, this effect perhaps also hinges on the increased use of corporations and the increase in the % of the economy channeled through indefinitely long-lived entities that are, ultimately, designed to moderate risk (premia).

We are replacing illiquid assets with liquid assets which means we can expect inflation. More liquidity, more inflation... In this sense QE is inflationary... But i agree with interest rate increase because interest cost will go up, prices will go -up. Our purpose to generate inflation will be solved. We want a policy which can make people spend more; it is a demand problem, even if it is under compulsion under price-rise. If inflation is high people will have to spend more. But if prices go down due to disinflation, probably deflation, people will spend willingly because now they have more money, they do not need wages/income to increase, when prices fall real wages increase. We do not need persistent deflation but a short-period of lower prices to remove excess supply and then back to the normal. The problem with QE is that we need to pursue it for a very long-time if we have to bring unemployment to its natural rate because we need to remove excess supply and we will have to wait till people’s income increase, which is constant in the short-run. Higher real wages are easy to achieve than a nominal appreciation during a recession hang-over.

Our other problem is liquidity-trap a situation which has made people accumulate reserve and post-pone spending because they are expecting a deflation and interest rate are at zero. To improve savings in banks we need to improve return on savings. We will not need QE to improve banks balance-sheet it will get repaired itself. More reserves for lending.

I think deflation is also a rational expectation because when prices are elevated after a period of increase people expect it to come down during recession. We should not go opposite of the popular-expectation…

Cute modeling, but about as appropriate as a bikini and lipstick on an 8-year old. No international cointegration, no drivers of inflation except the mob (oh excuse me, I meant the interest rate). The Fischer relation (not Stan) deserves better; but, if you're unwilling to raise wages, investment, and social spending, by all means raise interest, it is, after all a cost.

"One scenario that emerges in Steve's model is R =1 and K = L > 1. In this case, the economy is at the ZLB, but government liabilities (cash and bonds -- they are perfect substitutes in this case) exhibit a liquidity premium. "

If cash exhibits a liquidity premium that means it is providing investors with valuable liquidity services on the margin. Investors will only rent out this cash at some positive rate so that they can be compensated for foregone liquidity services. But the existence of a positive rate means we are not at the zero-lower bound. This is why the above phrase confuses me, since it seems to indicate we can simultaneously be at the ZLB and money can exhibit a liquidity premium -- but the existence of these two together is impossible. When money exhibits a liquidity premium, we are by definition off of the ZLB.

JP, maybe the confusion lies in the term "premium" -- premium relative to what?

Clearly, if R = 1, then money can have no liquidity premium relative to bonds, if both money and bonds are the same thing.

But money and bonds as an asset class can still be in "short supply," even if the nominal interest rate is zero. When this is the case, then the purchasing power of these assets will exceed their "fundamental" or "socially desirable" values.

"Clearly, if R = 1, then money can have no liquidity premium relative to bonds, if both money and bonds are the same thing."

I agree.

"But money and bonds as an asset class can still be in "short supply," even if the nominal interest rate is zero."

But if money is in short supply, then people will put a positive marginal value on the liquidity services it throws off. In which case, they require a positive rental rate in order to compensate them for forgoing those services overnight. Which means the overnight rate, or nominal interest rate, can't be zero. So when money is in short supply, nominal rates must be greater than zero. What am I not getting?

I'm not sure that's right; i.e. It doesn't seem like it's the real vs. the nominal rental rate that explains why a 0% rate is consistent with a positive convenience yield on bonds/money. Unless deflation is expected, a risk free asset which pays no interest with a positive liquidity yield should have a positive nominal rental rate -- and the model does not seem to require expected deflation to hit the zlb, right? But the financial frictions and safe asset shortage at the core of the model might explain it. The only way you can rent out money/bonds is to exchange it for a riskier IOU (by definition, after all if you collateralized your loan with Treasuries then the convenience yield to the borrower is eliminated). So there are plenty of willing borrowers at 0% when expected inflation => 0, but lenders would prefer to exploit the convenience yield themselves (even if the marginal benefit to the particular lender is below the hypothetical market convenience yield) than take the risky 1% it is worth to borrowers in convenience yield.

But what could explain the fact that there is any lending at 0% then? Maybe the only lending that takes place is infra-marginal lending among holders and borrowers satiated or near satiated in safe assets.

Consider two risk free assets, labeled "capital" and "money." Capital is illiquid. Money is liquid. Capital earns nominal (gross) interest R. Money earns nominal (gross) interest 1. If both assets are to be willingly held in individual wealth portfolios, then the following no arbitrage condition must hold: R = L, where L is the liquidity premium on money.

If we denote the (gross) rate of inflation by P, then the same no arbitrage condition implies: R/P = L/P.

Because capital is illiquid, it will earn the real rate of interest (G/B), so R/P = G/B (see equations above in my post).

Let's assume zero growth for now, so that G=1. Of course, B<1.

So we have R/P = 1/B, for the illiquid asset.

Of course, we also have L/P = 1/B for money, the liquid asset.

Therefore, the real rate of return on money is 1/P = 1/(BL), where L>1.

So here, we have the real rate of return on money is less than the real rate of return on capital. Both assets are risk free here. The difference in rates of return is attributable to due to the liquidity premium.

Notice that it is possible for the following to hold: R/P = 1/B = L/P with P>1 (positive inflation). All that is required for positive inflation is a high enough liquidity premium L. This is despite the fact that money earns zero nominal interest.

Maybe this comes down to a difference in defining the "nominal rental rate?"

Unless I'm really missing what you're getting at, I think everything you're saying is very standard but not a reason why the ZLB (as JP's question above means to define it) could co-exist with a liquidity premium and inflation. Say as a random case in your example, if the liquidity yield on money is 3%, inflation is 2% and the discount rate is 1%: I agree that money can pay zero nominal interest, generate a -2% real return, and a 1% total return including the 3% convenience yield. But the rental cost of money, the nominal interest rate, is definitely not zero. The rental cost of money is 3%. And as long as the liquidity premium and inflation are both positive, then the nominal rental rate (the nominal rate paid to borrow money) must be positive. The nominal rental rate in your model (as I believe JP means it) is not the nominal interest paid on money, it is R. And R (R-1 actually) cannot be zero if both L and P are > 1. I read his question as "how can the liquidity premium on the money/bond asset and inflation both exist in a world where the nominal interest rate is zero?"

"The nominal rental rate in your model (as I believe JP means it) is not the nominal interest paid on money, it is R."

Well, in the model I described, R is not the nominal interest rate on money. R is the nominal interest rate earned by an illiquid (non-monetary) asset (the nominal rate of return R compensates the holder of the asset for its illiquidity).

There is more than one interest rate in this economy. Are we getting closer?

I can think clearer using actual examples of interest rates. Say that new bonds are being issued with a yield of 0. There is a liquidity premium on bonds and central bank deposits.

Given this setup, the overnight fed funds rate will be greater than 0 since anyone who lends central bank deposits requires a return to compensate them for foregone liquidity services. Any sort of bank-issued government insured term deposit will also yield more than 0, since anyone who swaps central bank deposits or cash for a relatively illiquid term deposit requires a return to compensate for foregone liquidity. The repo rate, however, will be 0. Repo is a swap of central bank deposits for bonds. Since both provide the same liquidity services on the margin, neither party to the repo transaction require an interest yield to compensate for forgone liquidity services.

So at the zero-lower bound that you & Steve are describing, there will be some market rates that are at 0 and some that are greater than 0. I think the traditional zero-lower bound has all interest rates at 0, or at least the fed funds rate at 0.

"Well, in the model I described, R is not the nominal interest rate on money. R is the nominal interest rate earned by an illiquid (non-monetary) asset (the nominal rate of return R compensates the holder of the asset for its illiquidity).

There is more than one interest rate in this economy. Are we getting closer?"

Maybe? I know R is not the nominal interest rate paid on money, but I am saying that R must equal the nominal interest rate to borrow and lend money if the the illiquid capital asset is risk free per your assumption. The liquidity yield is earned by the borrower of money, so the nominal rental rate of money is the nominal rate paid on money plus the liquidity yield, which = R-1 = inflation plus the discount rate in your story. JP isn't asking how the nominal rate paid on money/bonds can be zero, he is asking how the rental rate of the money/bond asset can be zero, if L and P are both > 1. Do you agree that the nominal rental rate (borrowing rate) of money in your model should be positive as long as L and P are > 1?

"I know R is not the nominal interest rate paid on money, but I am saying that R must equal the nominal interest rate to borrow and lend money if the the illiquid capital asset is risk free per your assumption."

You realize that there is no unsecured lending/borrowing in Steve's model, right? (The limited commitment friction prevents this). Might this be the source of the confusion? (If not, let us continue to sort things out.)

Also, I'm getting a little confused with the terminology you are introducing, like "nominal rental rate." Can't we just stick with the generic no-abritrage-condition R(i)*L(i) = (1/B)*P for assets i = 1,2,...,N and go from there?

"You realize that there is no unsecured lending/borrowing in Steve's model, right? (The limited commitment friction prevents this). Might this be the source of the confusion? (If not, let us continue to sort things out.)"

Woo-hoo. We have gotten there. This is exactly what I was trying to say in my original response to JP, with one twist. You are explaining why the model simply excludes unsecured lending, while I was reading JP's question as asking how we should think about such a model when in the actual liquidity trap we do in fact have unsecured lending, such as in the interbank market, at/near the zero bound.

Well, unsecured lending occurs in reality even away from the ZLB. It is straightforward to accommodate the coexistence of unsecured credit and exchange media by introducing some limited amount of record keeping. In some models, a subset of agents or trades are assumed to be "monitored" (i.e., recorded). Then the failure of not repaying debt is punishable by ostracism, for example. I'm not sure whether incorporating unsecured lending in the model will overturn Steve's results. Of course, it is important that credit markets do not work perfectly.

Collateral does not facilitate "lending" or "credit". The vast majority of collateralized obligations are rated AAA/AA.

I don't think we understand this collateral issue well. At its core, converting long term safe (credit risk free) collateral to s.t. assets is a duration bet. Someone along the chain is going to make or lose money depending on the volatility of interest rates. Therefore, one could imagine that the use of "collateral" is related to rate volatility expectations. Enter Fed forward rate guidance, which has held down rate volatility for much of the past decade. This guidance coincided with the explosion in shadow bank assets.

"Safe assets" are not safe from a duration risk standpoint. I think this bears keeping in mind.

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